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<div class="title">Filtering using F-transform </div>  </div>
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<div class="textblock"><h1>Goal </h1>
<p>This tutorial demonstrates to you how to use F-transform for image filtering. You will see:</p>
<ul>
<li>basic theory behind,</li>
<li>illustration of different settings.</li>
</ul>
<h1>Fuzzy transform application </h1>
<p>As I shown in previous tutorial, F-transform is a tool of fuzzy mathematics highly usable in image processing. Let me rewrite the formula using kernel \(g\) introduced before as well:</p>
<p class="formulaDsp">
\[ F^0_{kl}=\frac{\sum_{x=0}^{2h+1}\sum_{y=0}^{2h+1} \iota_{kl}(x,y) g(x,y)}{\sum_{x=0}^{2h+1}\sum_{y=0}^{2h+1} g(x,y)}, \]
</p>
<p>where \(\iota_{kl} \subset I\) centered to pixel \((k \cdot h,l \cdot h)\) and \(g\) is a kernel. More details can be found in related papers.</p>
<h1>Code </h1>
<div class="fragment"><div class="line"><span class="comment">/* Sample - Filtering</span></div><div class="line"><span class="comment"> * Target is to apply filtering using F-transform</span></div><div class="line"><span class="comment"> * on the image &quot;input.png&quot;. Two different kernels</span></div><div class="line"><span class="comment"> * are used, where bigger radius (100 in this case)</span></div><div class="line"><span class="comment"> * means higher level of blurriness.</span></div><div class="line"><span class="comment"> *</span></div><div class="line"><span class="comment"> * Image &quot;output1_filter.png&quot; is created from &quot;input.png&quot;</span></div><div class="line"><span class="comment"> * using &quot;kernel1&quot; with radius 3.</span></div><div class="line"><span class="comment"> *</span></div><div class="line"><span class="comment"> * Image &quot;output2_filter.png&quot; is created from &quot;input.png&quot;</span></div><div class="line"><span class="comment"> * using &quot;kernel2&quot; with radius 100.</span></div><div class="line"><span class="comment"> *</span></div><div class="line"><span class="comment"> * Both kernels are created from linear function, using</span></div><div class="line"><span class="comment"> * linear interpolation (parameter ft:LINEAR).</span></div><div class="line"><span class="comment"> */</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &quot;<a class="code" href="../../d0/d9c/core_2include_2opencv2_2core_8hpp.html">opencv2/core.hpp</a>&quot;</span></div><div class="line"><span class="preprocessor">#include &quot;<a class="code" href="../../d4/dd5/highgui_8hpp.html">opencv2/highgui.hpp</a>&quot;</span></div><div class="line"><span class="preprocessor">#include &quot;<a class="code" href="../../df/dc0/fuzzy_8hpp.html">opencv2/fuzzy.hpp</a>&quot;</span></div><div class="line"></div><div class="line"><span class="keyword">using namespace </span>std;</div><div class="line"><span class="keyword">using namespace </span><a class="code" href="../../d2/d75/namespacecv.html">cv</a>;</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main(<span class="keywordtype">void</span>)</div><div class="line">{</div><div class="line">    <span class="comment">// Input image</span></div><div class="line">    <a class="code" href="../../d3/d63/classcv_1_1Mat.html">Mat</a> I = <a class="code" href="../../d4/da8/group__imgcodecs.html#ga288b8b3da0892bd651fce07b3bbd3a56">imread</a>(<span class="stringliteral">&quot;input.png&quot;</span>);</div><div class="line"></div><div class="line">    <span class="comment">// Kernel creation</span></div><div class="line">    <a class="code" href="../../d3/d63/classcv_1_1Mat.html">Mat</a> kernel1, kernel2;</div><div class="line"></div><div class="line">    <a class="code" href="../../d1/dfc/group__f__image.html#ga0e5694cf8dd380d512e915660870027c">ft::createKernel</a>(<a class="code" href="../../df/d5b/group__fuzzy.html#gga847dcdc873389a919e2c3ba910a2bd39a4d3d0c977941e929e23e9c6cb4ae4715">ft::LINEAR</a>, 3, kernel1, 3);</div><div class="line">    <a class="code" href="../../d1/dfc/group__f__image.html#ga0e5694cf8dd380d512e915660870027c">ft::createKernel</a>(<a class="code" href="../../df/d5b/group__fuzzy.html#gga847dcdc873389a919e2c3ba910a2bd39a4d3d0c977941e929e23e9c6cb4ae4715">ft::LINEAR</a>, 100, kernel2, 3);</div><div class="line"></div><div class="line">    <span class="comment">// Filtering</span></div><div class="line">    <a class="code" href="../../d3/d63/classcv_1_1Mat.html">Mat</a> output1, output2;</div><div class="line"></div><div class="line">    <a class="code" href="../../d1/dfc/group__f__image.html#ga99c17738a69b556b18bc86e7aee1a1d9">ft::filter</a>(I, kernel1, output1);</div><div class="line">    <a class="code" href="../../d1/dfc/group__f__image.html#ga99c17738a69b556b18bc86e7aee1a1d9">ft::filter</a>(I, kernel2, output2);</div><div class="line"></div><div class="line">    <span class="comment">// Save output</span></div><div class="line"></div><div class="line">    <a class="code" href="../../d4/da8/group__imgcodecs.html#gabbc7ef1aa2edfaa87772f1202d67e0ce">imwrite</a>(<span class="stringliteral">&quot;output1_filter.png&quot;</span>, output1);</div><div class="line">    <a class="code" href="../../d4/da8/group__imgcodecs.html#gabbc7ef1aa2edfaa87772f1202d67e0ce">imwrite</a>(<span class="stringliteral">&quot;output2_filter.png&quot;</span>, output2);</div><div class="line"></div><div class="line">    <span class="keywordflow">return</span> 0;</div><div class="line">}</div></div><!-- fragment --><h1>Explanation </h1>
<p>Image filtering changes input in a defined way to enhance or simply change some concrete feature. Let me demonstrate some simple blur.</p>
<p>As a first step, we load input image.</p>
<div class="fragment"><div class="line"><span class="comment">// Input image</span></div><div class="line">Mat I = <a class="code" href="../../d4/da8/group__imgcodecs.html#ga288b8b3da0892bd651fce07b3bbd3a56">imread</a>(<span class="stringliteral">&quot;input.png&quot;</span>);</div></div><!-- fragment --><p>Following the F-transform formula, we must specify a kernel.</p>
<div class="fragment"><div class="line"><span class="comment">// Kernel cretion</span></div><div class="line">Mat kernel1, kernel2;</div><div class="line"></div><div class="line"><a class="code" href="../../d1/dfc/group__f__image.html#ga0e5694cf8dd380d512e915660870027c">ft::createKernel</a>(<a class="code" href="../../df/d5b/group__fuzzy.html#gga847dcdc873389a919e2c3ba910a2bd39a4d3d0c977941e929e23e9c6cb4ae4715">ft::LINEAR</a>, 3, kernel1, 3);</div><div class="line"><a class="code" href="../../d1/dfc/group__f__image.html#ga0e5694cf8dd380d512e915660870027c">ft::createKernel</a>(<a class="code" href="../../df/d5b/group__fuzzy.html#gga847dcdc873389a919e2c3ba910a2bd39a4d3d0c977941e929e23e9c6cb4ae4715">ft::LINEAR</a>, 100, kernel2, 3);</div></div><!-- fragment --><blockquote class="doxtable">
<p>So now, we have two kernels that differ in <code>radius</code>. Bigger radius leads to bigger blur. </p>
</blockquote>
<p>The filtering itself is applied as shown below.</p>
<div class="fragment"><div class="line"><span class="comment">// Filtering</span></div><div class="line">Mat output1, output2;</div><div class="line"></div><div class="line"><a class="code" href="../../d1/dfc/group__f__image.html#ga99c17738a69b556b18bc86e7aee1a1d9">ft::filter</a>(I, kernel1, output1);</div><div class="line"><a class="code" href="../../d1/dfc/group__f__image.html#ga99c17738a69b556b18bc86e7aee1a1d9">ft::filter</a>(I, kernel2, output2);</div></div><!-- fragment --><p>Output images look as follows.</p>
<div class="image">
<img src="../../fuzzy_filt_output.jpg" alt="fuzzy_filt_output.jpg"/>
<div class="caption">
input, output1 (radius 3), output2 (radius 100)</div></div>
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